Euler and a Proof of the Functional Equation for the Riemann Zeta-Function He Could Have Given

We explain how Euler could have proved a functional equation, which is equivalent to the one for the Riemann zeta-function, that he conjectured in his paper { ``Remarques sur un beau rapport entre les series des puissances tant directes que reciproques"} {E352} (E352: ``Remarks on the beautiful relation between the series of the direct and reciprocal powers").